Probabilities of rare events in product kernel aggregation: An exact formula and phase diagram

TL;DR

This work provides an exact, finite-$M$ formulation for the probability of observing a given number of particles in product-kernel aggregation, enabling precise large-deviation analysis. It derives exponential moments exactly for integer $p$ and extends to real $p$ through a replica-based conjecture, yielding a closed form for the scaled cumulant generating function $oldsymbol{\lambda}( au,p)$ and its maximization structure. The authors uncover singularities in these moments that map to a rich phase diagram for the large-deviation function (LDF), including a tricritical point separating continuous and discontinuous regimes, and they compute the convex envelope of the LDF (CELDF). The approach links dynamic aggregation to static graph ensembles and provides a controlled route to asymptotics, with explicit results for small $oldsymbol{\phi}=N/M$ and potential for broad generalizations.

Abstract

We present an exact method for calculating the large deviation function describing rare fluctuations in the number of particles for product-kernel aggregation. Starting from the master equation, we derive an exact integral representation for the probability $P(M,N,t)$ of observing $N$ particles at time $t$ starting from $M$ monomers for any finite $M, N, t$. From this, we obtain an exact expression for the exponential moment $\langle p^N\rangle$ for integer $p$. Employing a replica conjecture -- numerically validated by finite-$M$ scaling -- we extend this result to real $p \geq 0$. The convex envelope of the large deviation function, obtained via a Legendre-Fenchel transform of the exponential moment, shows singular behavior. The singular structure allows us to construct the full phase diagram of product-kernel aggregation, which contains a tricritical point, separating continuous and discontinuous transitions. We also compute the asymptotic form of the LDF for small $N/M$.

Figures (12)

• Figure 1: LDF obtained from the integral representation of $P(M,N,t)$ given in Eq. \ref{['prob27']}for $M$ using when (a) $\tau=0.5$ and (b) $\tau=4.0$. Inset: $\delta$, the difference between the LDF for $M=800$ and its CELDF.
• Figure 2: The possible solutions of Eq. \ref{['crit']} for $x$, depending on the values of $\tau/Z$. (a) No solutions when $\tau/Z>e^{-1}$, (b) one solution $x^{(0)}=1$ when $\tau/Z=e^{-1}$ and (c) two distinct solutions $x^{(1)}$ and $x^{(2)}$ when $\tau/Z<e^{-1}$.
• Figure 3: Comparison of the exponential moment obtained from replica calculation (Eq. \ref{['replica']}) for infinite $M$ with the series expansion (Eq. \ref{['expmoment']}) for finite $M$, when (a) $p=2.5$, (d) $p=1.5$ and (g) $p=0.5$. The difference between the exponential moment obtained from series expansion and replica calculation, denoted by $\Delta \lambda$ are shown in (b) $p=2.5$, (e) $p=1.5$ and (h) $p=0.5$. The corresponding insets show the data collapse when plotted against $M \Delta \lambda$. The variation of $\zeta$ with $\tau$ for (c) $p=2.5$, (f) $p=1.5$ and (i) $p=0.5$. The transition time $\tau_c$ is identified as the smallest $\tau$ for which $\zeta$ is non-zero.
• Figure 4: Comparison of the transition time $\tau_c$, as given in Eq. \ref{['tauc']}, with the $\tau_c$ obtained from the numerical solution of $\zeta$ with $\tau$ as shown in Fig. \ref{['comparison']}(c), (f), (i).
• Figure 5: The variation of $\langle \phi \rangle_p$, the mean fraction of particles, with $p$, for different values of $\tau$.